The talk Vibrations of a lumped parameter mass-spring-dashpot system where in the spring is defined through an implicit constitutive relation, by Vít Průša, will be held on Monday April 13, 2015 at 8:30 in room K4. After that, Dalibor Pražák will present the talk Non-standard damped oscillators.

Abstract -- Vít Průša's talk

The standard setting concerning vibrations of lumped parameter systems is based on the assumption that the mechanical response of the elements of the system is given explicitly in terms of kinematical variables. In particular, the force in a spring element is assumed to be given as a function of the displacement from the equilibrium position. However, some simple mechanical systems such as linear springs with limited compressibility/extensibility do not fit into the standard setting. In this case the displacement must be written as a function the force. In general, the mechanical response of such elements must be described by an implicit relation between the force and kinematical variables.

We study the behaviour of a particular lumped parameter system whose mechanical response is given by a non-invertible expression for the displacement in terms of the force, under harmonic external force. We show that a solution to the original system wherein the displacement is given in terms of the force can be obtained as a limit of a sequence of approximate problems. The approximate problems are designed in such a way that they can be solved using standard numerical methods, and one can avoid using concepts such as set valued mappings. Moreover, we show that the ''bounce back'' behaviour of the system with linear spring with limited compressibility/extensibility is a direct consequence of the assumed constitutive relation. There is no need to \emph{a priori} supply the rules for the bounce back (impact rules). Further, we show that the advocated approximation procedure is capable of describing the behaviour of lumped parameter system even in the situations where the governing ordinary differential equation collapses to an algebraic equation. The analytical results are demonstrated by a numerical experiment.

The damped oscillators of the form (1) x'' + a(x)x' + b(x) = f(t) are classical models in mechanics and for regular enough a(.), b(.), say C^1 or Lipschitz, the mathematical theory is very well understood.

Non-standard analysis (NSA), on the other hand, is a rather strong and abstract logical framework, using which various mathematical theories can be embedded into richer universes with non-standard ("ideal") elements. The simplest and most famous example are infinitely large and small numbers (which are thought by some advocates of NSA to be fatally missing from Calculus for more nearly 200 years by now.)

Curiously enough, some nonstandard choices of the functions a(.) and b(.), taking infinitely large values, or with infinitely steep growth, are natural models of some "non-standard" mechanical elements: damper with Coulomb's friction, inextensible string, or more generally, collision of a moving mass with a wall.

In our talk, we will see how these situations can be modelled within the framework of NSA. We show that interesting dynamics occurs and even more, new interesting questions can be asked.